The collected mathematical papers of Arthur Cayley.

407]

CURVES WHICH SATISFY GIVEN CONDITIONS.

281

127. Referring to Nos. 41 to 47 of the First Memoir, for convenience I collect the
capitals which belong to a single curve, giving the values in terms of m, n, a as follows.

(1, 1, 1, 1)
A =18(8-1)

(1)
(2)
(4)
(3)

= I m4
-9

B = 8(n-4)(m-4) =

In3k
2

- -m3 + 2m2n - _m2 - 2mn + 8n2 + rm - 2n
+ oa(- m22 + 3m - 6. + 4) + a2;
-2m3 - nm2n + 4mn2 + l0n2- 14mn- 16n2- 8m3 + 64n
+a( - -mn + 6m+ 6n - 24);
-lm2n2+ 2m3 - m2n - m_2- 18nm2 + mn + 5n2 + 40m - 5n
+a(-tnz2 +-27m -1?);
4I -15);
-m3 + +Lm2 -18m
+a( o nf 2 - rnm + 6).

C = T. 2 (m-4)(m-5)=
D = =. - (.n - 3)(m,- 4) =

(2, 1, 1)
E = (n-4)

(3)
(3)
(6)
(2)

21 2

F =28(m-3)

m3

- 2m2 -+a(
- 4m2 +
n2+ 8m(
mn + 8a ( -
+a(

rmn + 4?n2 + 2m - 16a
-n + 6);
8mn + 3m-24n
- 3m + 9);
mn - 4n2 - 32m + 4n
- 3m +12);

G =2(m - 4)

-) [==. (in - 3) (n3) - 4) szuprc];

(1)
(2)
(5)

H = SK

I = (n-3)(m-4)

+ a ( 2
- 3m2(
+a(
- 3m92/

+ 3mn-12n2
- im+ -V-n ) - 32;
+ 9mn n+ 12n2 - 36n
mn -3m- 4n + 12);
+ 9m + a (m - 3).

J =t (m-3)

(2, 2)
(9) K =r
(3) L = /(rT-3)
(1) M = -/e (, - 1)
(2) N=
(1) 0 =
C. VI.

n2 + 4nm - In + a ( - 3);
= - 3n2 + 9n + a( I - 3);
= -2 + n + a(-3n —)+ 2 2;
- 3m +a( 1);
= - 3n + a ( 1).

36